/ 
Math-PlanePath-127
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::QuintetCurve

NAME

Math::PlanePath::QuintetCurve -- self-similar "plus" shaped curve

SYNOPSIS

use Math::PlanePath::QuintetCurve;
my $path = Math::PlanePath::QuintetCurve->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path is Mandelbrot's "quartet" trace of spiralling self-similar "+" shape,

        125--...                 93--92                      11
          |                       |   |
    123-124                      94  91--90--89--88          10
      |                           |               |
    122-121-120 103-102          95  82--83  86--87           9
              |   |   |           |   |   |   |
    115-116 119 104 101-100--99  96  81  84--85               8
      |   |   |   |           |   |   |
113-114 117-118 105  32--33  98--97  80--79--78               7
  |               |   |   |                   |
112-111-110-109 106  31  34--35--36--37  76--77               6
              |   |   |               |   |
            108-107  30  43--42  39--38  75                   5
                      |   |   |   |       |
             25--26  29  44  41--40  73--74                   4
              |   |   |   |           |
         23--24  27--28  45--46--47  72--71--70--69--68       3
          |                       |                   |
         22--21--20--19--18  49--48  55--56--57  66--67       2
                          |   |       |       |   |
          5---6---7  16--17  50--51  54  59--58  65           1
          |       |   |           |   |   |       |
  0---1   4   9---8  15          52--53  60--61  64       <- Y=0
      |   |   |       |                       |   |
      2---3  10--11  14                      62--63          -1
                  |   |
                 12--13                                      -2

  ^
 X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 ...

As per

Benoit B. Mandelbrot, "The Fractal Geometry of Nature", W. H. Freeman
and Co., 1983, ISBN 0-7167-1186-9, section 7, "Harnessing the Peano
Monster Curves", pages 72-73.

Mandelbrot calls this a "quartet", taken as 4 parts around a further middle part (like 4 players around a table). The module name "quintet" here is a mistake, though it does suggest the base-5 nature of the curve.

The base figure is the initial N=0 to N=5.

        5
        |
        |
0---1   4      base figure
    |   |
    |   |
    2---3

It corresponds to a traversal of the following "+" shape,

     .... 5
     .    |
     .   <|
          |
0----1 .. 4 ....
  v  |    |    .
.    |>   |>   .
.    |    |    .
.... 2----3 ....
     . v  .
     .    .
     .    .
     . .. .

The "v" and ">" notches are the side the figure is directed at the higher replications. The 0, 2 and 3 sub-curves are the right hand side of the line and are a plain repetition of the base figure. The 1 and 4 parts are to the left and are a reversal. The first such reversal is seen in the sample above as N=5 to N=10. ..... . .

5---6---7 ...
.   .   |   .
.       |   .   reversed figure
... 9---8 ...
    |   .
    |   .
   10 ...

Mandelbrot gives the expansion without designating start and end. The start is chosen here so the expansion has sub-curve 0 forward (not reverse). This ensures the next expansion has the curve the same up to the preceding level, and extending from there.

In the base figure it can be seen the N=5 endpoint is rotated up around from the N=0 to N=1 direction. This makes successive higher levels slowly spiral around.

base b = 2 + i
N = 5^level
angle = level * arg(b) = level*atan(1/2)
      = level * 26.56 degrees

In the sample shown above N=125 is level=3 and has spiralled around to angle 3*26.56=79.7 degrees. The next level goes to X negative in the second quadrant. A full circle around the plane is approximately level 14.

Arms

The optional arms => $a parameter can give 1 to 4 copies of the curve, each advancing successively. For example arms=>4 is as follows. N=4*k points are the plain curve, and N=4*k+1, N=4*k+2 and N=4*k+3 are rotated copies of it.

                69--65                      ...
                 |   |                       |
..-117-113-109  73  61--57--53--49         120
             |   |               |           |
       101-105  77  25--29  41--45 100-104 116
         |       |   |   |   |       |   |   |
        97--93  81  21  33--37  92--96 108-112
             |   |   |           |
    50--46  89--85  17--13-- 9  88--84--80--76--72
     |   |                   |                   |
    54  42--38  10-- 6   1-- 5  20--24--28  64--68
     |       |   |   |           |       |   |
    58  30--34  14   2   0-- 4  16  36--32  60
     |   |       |           |   |   |       |
66--62  26--22--18   7-- 3   8--12  40--44  56
 |                   |                   |   |
70--74--78--82--86  11--15--19  87--91  48--52
                 |           |   |   |
   110-106  94--90  39--35  23  83  95--99
     |   |   |       |   |   |   |       |
   114 102--98  47--43  31--27  79 107-103
     |           |               |   |
   118          51--55--59--63  75 111-115-119-..
     |                       |   |
    ...                     67--71

The curve is essentially an ever expanding "+" shape with one corner at the origin. Four such shapes pack as follows,

            +---+
            |   |
    +---@---    +---+
    |   |     B     |
+---+   +---+   +---@
|     C     |   |   |
+---+   +---O---+   +---+
    |   |   |     A     |
    @---+   +---+   +---+
    |     D     |   |
    +---+   +---@---+ 
        |   |
        +---+

At higher replication levels the sides are wiggly and spiralling and the centres of each rotate around, but their sides are symmetric and mesh together perfectly to fill the plane.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::QuintetCurve->new ()
$path = Math::PlanePath::QuintetCurve->new (arms => $a)

Create and return a new path object.

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

Fractional positions give an X,Y position along a straight line between the integer positions.

$n = $path->n_start()

Return 0, the first N in the path.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

In the current code the returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle, but don't rely on that yet since finding the exact range is a touch on the slow side. (The advantage of which though is that it helps avoid very big ranges from a simple over-estimate.)

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, 5**$level), or for multiple arms return (0, $arms * 5**$level).

There are 5^level + 1 points in a level, numbered starting from 0. On the second and subsequent arms the origin is omitted (so as not to repeat that point) and so just 5^level for them, giving 5^level+1 + (arms-1)*5^level = arms*5^level + 1 many points starting from 0.

FORMULAS

X,Y to N

The current approach uses the QuintetCentres xy_to_n(). Because the tiling in QuintetCurve and QuintetCentres is the same, the X,Y coordinates for a given N are no more than 1 away in the grid.

The way the two lowest shapes are arranged in fact means that for a QuintetCurve N at X,Y then the same N on the QuintetCentres is at one of three locations

X, Y          same
X, Y+1        up
X-1, Y+1      up and left
X-1, Y        left

This is so even when the "arms" multiple paths are in use (the same arms in both coordinates).

Is there an easy way to know which of the four offsets is right? The current approach is to give each to QuintetCentres to make an N, put that N back through n_to_xy() to see if it's the target $n.

SEE ALSO

Math::PlanePath, Math::PlanePath::QuintetCentres, Math::PlanePath::QuintetReplicate, Math::PlanePath::Flowsnake

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.